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The SIS epidemic model with Markovian switching. (English) Zbl 1271.92030
This paper first examined a SIS epidemic model with Markovian switching of two states and presented its explicit solution. This paper also established the conditions for extinction and persistence and compared them with the corresponding conditions for the deterministic SIS epidemic models. Then the paper extended these results for two-state Markovian chains to a finite-state Markovian chain. The paper contains many useful probabilistic computations.
Reviewer: Fuke Wu (Wuhan)

92D30 Epidemiology
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
92-08 Computational methods for problems pertaining to biology
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[1] Du, N.H.; Kon, R.; Sato, K.; Takeuchi, Y., Dynamical behaviour of lotka – volterra competition systems: non autonomous bistable case and the effect of telegraph noise, J. comput. appl. math., 170, 399-422, (2004) · Zbl 1089.34047
[2] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht · Zbl 0752.34039
[3] Mao, X., Stability of stochastic differential equations with respect to semimartingales, (1991), Longman Scientific and Technical London · Zbl 0724.60059
[4] Mao, X., Exponential stability of stochastic differential equations, (1994), Marcel Dekker New York · Zbl 0851.93074
[5] Mao, X., Stochastic differential equations and applications, (2007), Horwood Publishing Chichester
[6] Slatkin, M., The dynamics of a population in a Markovian environment, Ecology, 59, 249-256, (1978)
[7] Takeuchi, Y.; Du, N.H.; Hieu, N.T.; Sato, K., Evolution of predator – prey systems described by a lotka – volterra equation under random environment, J. math. anal. appl., 323, 938-957, (2006) · Zbl 1113.34042
[8] Gilpin, M.E., Predator-prey communities, (1975), Princeton University Press Princeton
[9] Takeuchi, Y., Global dynamical properties of lotka – volterra systems, (1996), World Scientific Publishing Company Singapore · Zbl 0844.34006
[10] Padilla, D.K.; Adolph, S.C., Plastic inducible morphologies are not always adaptive: the importance of time delays in a stochastic environment, Evol. ecol., 10, 105-117, (1996)
[11] Anderson, D.R., Optimal exploitation strategies for an animal population in a Markovian environment: a theory and an example, Ecology, 56, 1281-1297, (1975)
[12] Peccoud, J.; Ycart, B., Markovian modeling of gene-product synthesis, Theoret. pop. biol., 48, 2, 222-234, (1995) · Zbl 0865.92006
[13] Caswell, H.; Cohen, J.E., Red, white and blue: environmental variance spectra and coexistence in metapopulations, J. theoret. biol., 176, 301-316, (1995)
[14] Hethcote, H.W., Qualitative analyses of communicable disease models, Math. biosci., 28, 335-356, (1976) · Zbl 0326.92017
[15] Hethcote, H.W.; Yorke, J.A., ()
[16] Lamb, K.E.; Greenhalgh, D.; Robertson, C., A simple mathematical model for genetic effects in pneumococcal carriage and transmission, J. comput. appl. math., 235, 7, 1812-1818, (2010) · Zbl 1207.92025
[17] Lipsitch, M., Vaccination against colonizing bacteria with multiple serotypes, Proc. natl. acad. sci., 94, 6571-6576, (1997)
[18] Brauer, F.; Allen, L.J.S.; Van den Driessche, P.; Wu, J., ()
[19] Iannelli, M.; Milner, F.A.; Pugliese, A., Analytical and numerical results for the age-structured SIS epidemic model with mixed inter-intracohort transmission, SIAM J. math. anal., 23, 3, 662-688, (1992) · Zbl 0776.35032
[20] Feng, Z.; Huang, W.; Castillo-Chavez, C., Global behaviour of a multi-group SIS epidemic model with age-structure, J. differential equations, 218, 2, 292-324, (2005) · Zbl 1083.35020
[21] Neal, P., Stochastic and deterministic analysis of SIS household epidemics, Adv. appl. probab., 38, 4, 943-968, (2006) · Zbl 1103.92035
[22] Neal, P., The SIS great circle epidemic model, J. appl. prob., 45, 2, 513-530, (2008) · Zbl 1145.92030
[23] Li, J.; Ma, Z.; Zhou, Y., Global analysis of an SIS epidemic model with a simple vaccination and multiple endemic equilibria, Acta math. sci., 26, 83-93, (2006) · Zbl 1092.92041
[24] Van den Driessche, P.; Watmough, J., A simple SIS epidemic model with backward bifurcation, J. math. biol., 40, 525-540, (2000) · Zbl 0961.92029
[25] Andersson, P.; Lindenstrand, D., A stochastic SIS epidemic with demography: initial stages and time to extinction, J. math. biol., 62, 333-348, (2011) · Zbl 1232.92062
[26] Gray, A.; Greenhalgh, D.; Hu, L.; Mao, X.; Pan, J., A stochastic differential equation SIS epidemic model, SIAM J. appl. math., 71, 876-902, (2011) · Zbl 1263.34068
[27] Yang, Q.; Jiang, D.; Shi, N.; Ji, C., The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. math. anal. appl., 388, 248-271, (2012) · Zbl 1231.92058
[28] Liu, X.; Stechlinski, P., Pulse and constant control schemes for epidemic models with seasonality, Nonlinear anal. real world appl., 12, 931-946, (2011) · Zbl 1203.92058
[29] Bhattacharyya, R.; Mukhopadhyay, B., On an eco-epidemiological model with prey harvesting and predator switching: local and global perspectives, Nonlinear anal. real world appl., 11, 3824-3833, (2010) · Zbl 1205.34048
[30] Artalejo, J.R.; Economou, A.; Lopez-Herrero, M.J., On the number of recovered individuals in the SIS and SIR stochastic epidemic models, Math. biosci., 228, 45-55, (2010) · Zbl 1200.92035
[31] Anderson, W.J., Continuous-time Markov chains, (1991), Springer-Verlag Berlin-Heidelberg · Zbl 0721.60081
[32] Diekmann, O.; Heesterbeek, J.A.P., Mathematical epidemiology of infectious diseases: model building, analysis and interpretation, (2000), John Wiley Chichester · Zbl 0997.92505
[33] Mao, X., Stability of stochastic differential equations with Markovian switching, Stochastic process. appl., 79, 45-67, (1999) · Zbl 0962.60043
[34] Mao, X.; Yuan, C., Stochastic differential equations with Markovian switching, (2006), Imperial College Press London · Zbl 1126.60002
[35] Brugger, S.D.; Hathaway, L.J.; M├╝hlemann, K., Detection of streptococcus pneumoniae strain cocolonization in the nasopharynx, J. clin. microbiol., 47, 6, 1750-1756, (2009)
[36] Coffey, T.J.; Enright, M.C.; Daniels, M.; Morona, J.K.; Morona, R.; Hryniewicz, W.; Paton, J.C.; Spratt, B.G., Recombinational exchanges at the capsular polysaccharide biosynthetic locus lead to frequent serotype changes among natural isolates of streptococcus pneumoniae, Mol. microbiol., 27, 73-83, (1998)
[37] A. Weir, Modelling the impact of vaccination and competition on pneumococcal carriage and disease in Scotland, unpublished Ph.D. Thesis, University of Strathclyde, Glasgow, Scotland, 2009.
[38] P. Farrington, What is the reproduction number for pneumococcal infection, and does it matter? in: 4th International Symposium on Pneumococci and Pneumococcal Diseases, May 9-13 2004 at Marina Congress Center, Helsinki, Finland, 2004.
[39] Zhang, Q.; Arnaoutakis, K.; Murdoch, C.; Lakshman, R.; Race, G.; Burkinshaw, R.; Finn, A., Mucosal immune responses to capsular pneumococcal polysaccharides in immunized preschool children and controls with similar nasal pneumococcal colonization rates, Pediatr. infect. dis. J., 23, 307-313, (2004)
[40] Hoti, F.; Erasto, P.; Leino, T.; Auronen, K., Outbreaks of streptoccocus pneumoniae in day care cohorts in Finland - implications for elimination of transmission, BMC infectious diseases, 9, 102, (2009)
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